Slow biking and/or getting Uber pick-up if the weather really sucks should take about same amount of time, so I would recommend this as your best overall lowest expense/hassle/sweat option. I keep considering the possibility of a scooter, but that is only because I ride through some rough neighborhoods.

I would use one of your current bikes for the first 4 weeks of the job. Mix in walking. You start employment at a warm, convenient, easy time of the year for weather to commute while learning and building skills. After some time at the job, you might change methods. You could meet coworker to network and commute. You might find more optimal housing. No need to spend $$$ before you fully know the system. As you master the summer season, the appropriate answer for winter will take shape.

Without my bike commute, I would (and have in the past) turned into a potato. With the commute, each day is the better for the moderate metabolic zinger it gives me. Go bike. Possibly with one of your current bikes, and reconsider come autumn.

+1 for slow biking. Getting electric assist for 3 miles seems like overkill for someone who is already accustomed to riding regularly, even on a hilly route. Even if it's not the easiest method of getting to work, I'd choose the 30 minute cycle with a shower at the office over an hour walk. Its the most fun, and slightly quicker than walking, albeit more expensive.

I'm wearing business casual at the office, and bring in a fresh shirt rolled up around underwear. The shirt can get slightly creased during the joruney, but they will fall out within a few minutes of wearing due to body heat. I leave shoes and trousers and a towels at work, rotating them out for washing as needed.

Do you have a car sharing scheme at your new employers? I managed to do a car share where the other person was always the driver because my place was en route. This could be only for the more inclement weather and walking sounds fine for the other occasions. Three miles is a nice but not onerous walk. You may wish to do shopping and other errands on the return route. (Written as someone who never bikes).